You may be able to apply for the right to remain in the UK based on a human rights argument, namely your right to family and/or private life in the UK.
The right to family and private life is known as an Article 8 right, which derives from Article 8 of the European Convention on Human Rights (ECHR), which is part of UK law through the Human Rights Act. This right continues to exist after Brexit.
These applications can be based on a relationship between husband and wife, parent and child or indeed any other relationships that includes a level of dependency. It can also be based on requiring medical treatment or your length of residence in the UK (although living in the UK does not, in itself, amount to private life in this legal sense).
It is not easy to define what may and may not constitute a human right. But certainly one would argue that compelling and compassionate circumstances should exist. The definition is case-specific and is shaped largely by case law.
Human Rights is not an absolute right and there is an important balancing act that the Government must exercise in order for affective immigration control. The onus will be on you to demonstrate that your Article 8 rights outweigh the Governments. In essence case law dictates that there must be a level of proportionality when assessing whether your Article 8 rights will be breached.
There is significant case law that may protect your rights in the UK and allow you to maintain your relationships with loved ones. For example if you have minor children living in the UK then there is extensive case law and legislation in favour of protecting the bests interests of the child.
Get in touch with IJN Law if you think you can apply under Human Rights grounds.